can u please explain how the underlined differentiated part came:

Question 24:

If x=a sin 2t (1 + cos 2t) and y = b cos 2t (1-cos 2t), show that at  t = π 4 , d y d x , b a t = π 4 , d y d x = b a

Dear student,

There is mistake in the solution 

On differentiating (applying product rule)x=asin2t1+cos2tdxdt=1+cos2tddtasin2t+asin2tddt1+cos2tdxdt=1+cos2t×2acos2t+asin2t0-2sin2tdxdt=2acos2t1+cos2t-2asin22tdxdt=2acos2t+2acos22t-2asin22tSimilarly, y=bcos2t1-cos2tOn differentiatingdydt=ddtbcos2t1-cos2tdydt=bcos2tddt1-cos2t+1-cos2tddtbcos2t=bcos2t0+2sin2t+1-cos2t-2bsin2tdydt=2bsin2tcos2t-2bsin2t+2bsin2tcos2tdydt=bsin4t-2bsin2t+bsin4tdydx=bsin4t-2bsin2t+bsin4t2acos2t+2acos22t-2asin22tdydxt=π4=bsin4×π4-2bsin2×π4+bsin4×π42acos2×π4+2acos22×π4-2asin22×π4=bsinπ-2bsinπ2+bsinπ2acosπ2+2acos2π2-2asin2π2=0-2b+00+0-2a                      --[as sinπ=0, sinπ2=1 and cosπ2=0]=-2b-2adydxt=π4=baHence proved


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